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Global in Time Vortex Configurations for the $2$D Euler Equations

Juan Dávila, Manuel del Pino, Monica Musso, Shrish Parmeshwar

Abstract

We consider the problem of finding a solution to the incompressible Euler equations $$ ω_t + v\cdot \nabla ω= 0 \quad \hbox{ in } \mathbb{R}^2 \times (0,\infty), \quad v(x,t) = \frac 1{2π} \int_{{\mathbb R}^2} \frac {(y-x)^\perp}{|y-x|^2} ω(y,t)\, dy $$ that is close to a superposition of traveling vortices as $t\to \infty$. We employ a constructive approach by gluing classical traveling waves: two vortex-antivortex pairs traveling at main order with constant speed in opposite directions. More precisely, we find an initial condition that leads to a 4-vortex solution of the form $$ ω(x,t) = ω_0(x-ct\, e ) - ω_0 ( x+ ct \, e) + o(1) \ \hbox{ as } t\to\infty $$ where $$ ω_0( x ) = \frac 1{\varepsilon^{2}} W \left ( \frac {x-q} \varepsilon \right ) - \frac 1{\varepsilon^{2}}W \left ( \frac {x+q} \varepsilon \right ) + o(1) \ \hbox{ as } \varepsilon \to 0 $$ and $W(y)$ is a certain fixed smooth profile, radially symmetric, positive in the unit disc zero outside.

Global in Time Vortex Configurations for the $2$D Euler Equations

Abstract

We consider the problem of finding a solution to the incompressible Euler equations that is close to a superposition of traveling vortices as . We employ a constructive approach by gluing classical traveling waves: two vortex-antivortex pairs traveling at main order with constant speed in opposite directions. More precisely, we find an initial condition that leads to a 4-vortex solution of the form where and is a certain fixed smooth profile, radially symmetric, positive in the unit disc zero outside.
Paper Structure (21 sections, 23 theorems, 489 equations)

This paper contains 21 sections, 23 theorems, 489 equations.

Table of Contents

  1. Introduction
  2. Scheme of the proof
  3. Travelling vortex pair
  4. Modified vortex trajectories
  5. Approximate solution on $[T_0,T]$
  6. Construction of a solution
  7. Point Vortex Trajectories for Two Vortex Pairs
  8. First Approximation on $[T_{0},T]$
  9. Constructing Solutions on $[T_{0},T]$
  10. Homotopic Operators
  11. Weighted $L^{2}$ Function Space
  12. The Poisson Equation
  13. Lower Bound on a Quadratic Form
  14. Weighted $L^{2}$ A Priori Estimates
  15. A Priori Estimates on a Projected Linear Problem
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\gamma>18$, $q_\infty >0$. Then for all $T_0>0$ large enough satisfying K-T0-choices, there is a constant $C>0$ such that for all $\varepsilon >0$ small enough, there exists a solution to 2d-euler-vorticity-stream, $(\omega_\varepsilon , \Psi_\varepsilon)$, on the whole interval $[T_0,\infty)$, where the point $\left(p_{*}(t),q_{*}(t)\right)$ is defined in ppair2, and $p_{re}(t)$ and $q_{re}(

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 3.1
  • Remark 4.1
  • Lemma 4.2
  • ...and 36 more